All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$4.00$ each for students, and the group paid $$62.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$22.50$ each for teachers and $$7.50$ each for students, and the group paid $$150.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4y = 62}$ ${22.5x+7.5y = 150}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-12y = -186}$ ${22.5x+7.5y = 150}$ Add the top and bottom equations together. $ -4.5y = -36 $ $ y = \dfrac{-36}{-4.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {7.5x+4y = 62}$ to find $x$ ${7.5x + 4}{(8)}{= 62}$ $7.5x+32 = 62$ $7.5x = 30$ $x = \dfrac{30}{7.5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {22.5x+7.5y = 150}$ and get the same answer for $x$ ${22.5x + 7.5}{(8)}{= 150}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.